Finite Group Databasebeta

← #25749

Subgroups — #25749

Order 256.

826 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3498C211yesmaximal, characteristic, complemented
2#3153C211yesmaximal, characteristic, complemented
3#3040C211yesmaximal, characteristic, complemented
4C2×#455C212yesmaximal, complemented
5C2×#453C212yesmaximal, complemented
6#2657C211yesmaximal, characteristic, complemented
7#2004C212yesmaximal, complemented
8#2002C212yesmaximal, complemented
9#1965C211yesmaximal, characteristic, complemented
10#1963C211yesmaximal, characteristic, complemented
11#1759C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)C2212yes—
13C2×#137C2211yescharacteristic
14C2×#137C2212yescomplemented
15C2×AGL(1,ℤ/8)C2211yescharacteristic
16C2×AGL(1,ℤ/8)C2212yescomplemented
17#550—22no—
18D42—24no—
19D42—22no—
20#535—22no—
21#535C2212yes—
22#532—21no—
23#532C2211yescharacteristic
24C2×#127—21no—
25C2×#127C2211yescharacteristic
26C2×#127C2212yescomplemented
27C2×#124C2212yescomplemented
28C2×#124C2211yescharacteristic, complemented
29C4×#41C2211yescharacteristic
30#497—21no—
31#496—21no—
32#495—22no—
33#464—22no—
34#463—22no—
35#455—28no—
36#453—28no—
37#423C2211yescharacteristic, complemented
38C2×#104C2212yes—
39C2×#104C2212yescomplemented
40C2×#104C2212yescomplemented
41#419C2211yescharacteristic, complemented
42#418C2211yescharacteristic, complemented
43C2×#100C2212yescomplemented
44C2×#100C2212yescomplemented
45C4×M16C2211yescharacteristic, complemented
46#392—24no—
47#376C2211yescharacteristic, complemented
48#339C2212yescomplemented
49H(5,𝔽2)—24no—
50H(5,𝔽2)—24no—
51H(5,𝔽2)—42no—
52H(5,𝔽2)—42no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41C2312yes—
56C2×#41C2311yescharacteristic
57C22×Q8C2311yescharacteristic
58C22×D4—24no—
59C22×D4—24no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4—24no—
63C22×D4D414yes—
64C22×D4C2311yescharacteristic
65C22×D4C2312yes—
66#137—24no—
67#137—24no—
68#137—42no—
69AGL(1,ℤ/8)—24no—
70AGL(1,ℤ/8)—24no—
71AGL(1,ℤ/8)—42no—
72C2×Q16—21no—
73C2×Q16—22no—
74C2×SD16—21no—
75C2×SD16—22no—
76C2×SD16—21no—
77C2×SD16—22no—
78C2×D8—21no—
79C2×D8—22no—
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C2×M16C2312yes—
83C4⋊Q8D411yescharacteristic, complemented
84#127—24no—
85#127—22no—
86#127—21no—
87#127—21no—
88#127—22no—
89#127—24no—
90#127—42no—
91#127D411yescharacteristic, complemented
92#124—22no—
93#124—22no—
94#124—21no—
95#124—21no—
96#124—24no—
97#124—42no—
98#124D412yescomplemented
99#121—24no—
100#121—28no—
101#121—22no—
102#121—24no—
103#120—28no—
104#120—24no—
105C4×Q8D412yescomplemented
106C4×D4—24no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4D412yescomplemented
110#117—21no—
111#117C2311yescharacteristic
112C2×C22⋊C4—24no—
113C2×C22⋊C4—22no—
114C2×C42—21no—
115C2×C42C2312yes—
116C2×C42C2311yescharacteristic
117C4⋊C8—22no—
118#104—24no—
119#104—24no—
120#104D418yescomplemented
121#103—22no—
122#102—22no—
123#100—24no—
124#100—24no—
125#97—21no—
126C4×C8—21no—
127#95—22no—
128C24—24no—
129C24—21no—
130C24C2×D414yes—
131#41—22no—
132#41—22no—
133#41—22no—
134#41—22no—
135#41—22no—
136#41—44no—
137#41—44no—
138#41—44no—
139#41—44no—
140#41C2×D414yes—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C2×D412yes—
146C2×Q8C2×D411yescharacteristic
147C2×D4—21no—
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—22no—
151C2×D4—28no—
152C2×D4—24no—
153C2×D4—22no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—48no—
168C2×D4—48no—
169C2×D4—48no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—88no—
173C2×D4—84no—
174C2×D4—88no—
175C2×D4C2×D412yes—
176C2×D4C2×D411yescharacteristic
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4C2×D412yes—
184C22×C4C2411yescharacteristic
185Q16—44no—
186Q16—42no—
187SD16—44no—
188SD16—44no—
189SD16—42no—
190SD16—42no—
191D8—44no—
192D8—42no—
193M16—22no—
194M16—22no—
195M16—22no—
196M16C2×D414yescomplemented
197C2×C8—22no—
198C2×C8—21no—
199C2×C8—21no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4C2×D412yes—
203C22⋊C4—28no—
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—48no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C42—22no—
211C42—21no—
212C42—22no—
213C42C2×D411yescharacteristic
214C42C2×D411yescharacteristic
215C42C2×D412yes—
216C42C2×D414yescomplemented
217C23—21no—
218C23—24no—
219C23—22no—
220C23—24no—
221C23—24no—
222C23—48no—
223C23—48no—
224C23—44no—
225C23—41no—
226C23—42no—
227C23—88no—
228C23C22×D411yescharacteristic
229Q8—21no—
230Q8—22no—
231Q8—22no—
232Q8—41no—
233Q8—41no—
234Q8—42no—
235Q8#12712yes—
236D4—24no—
237D4—21no—
238D4—22no—
239D4—22no—
240D4—48no—
241D4—48no—
242D4—44no—
243D4—48no—
244D4—42no—
245D4—42no—
246D4—42no—
247D4—41no—
248D4—42no—
249D4—41no—
250D4—42no—
251D4—42no—
252D4—42no—
253D4—84no—
254D4—88no—
255D4—816no—
256D4#12712yes—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4#12012yes—
271C2×C4#12012yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D412yes—
274C2×C4C22×D411yescharacteristic
275C2×C4H(5,𝔽2)12yes—
276C8—24no—
277C8—41no—
278C8—41no—
279C22—22no—
280C22—22no—
281C22—42no—
282C22—42no—
283C22—44no—
284C22—48no—
285C22—44no—
286C22—84no—
287C22—84no—
288C22—84no—
289C22—81no—
290C22—82no—
291C22—168no—
292C22C2×#12011yescharacteristic
293C22D4212yes—
294C4—22no—
295C4—22no—
296C4—22no—
297C4—44no—
298C4—42no—
299C4—42no—
300C4—81no—
301C4C2×#12711yescharacteristic
302C4#53511yescharacteristic
303C4D4212yes—
304C2—22no—
305C2—42no—
306C2—84no—
307C2—82no—
308C2—161no—
309C2#243311yescharacteristic
310C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3498C2semidirect
2. #3153C2semidirect
3. #3040C2semidirect
4. C2×#455C2semidirect
5. C2×#453C2semidirect
6. #2657C2semidirect
7. #2004C2semidirect
8. #2002C2semidirect
9. #1965C2semidirect
10. #1963C2semidirect
11. #1759C2semidirect
14. C2×#137C22semidirect
16. C2×AGL(1,ℤ/8)C22semidirect
26. C2×#127C22semidirect
27. C2×#124C22semidirect
28. C2×#124C22semidirect
37. #423C22semidirect
39. C2×#104C22semidirect
40. C2×#104C22semidirect
41. #419C22semidirect
42. #418C22semidirect
43. C2×#100C22semidirect
44. C2×#100C22semidirect
45. C4×M16C22semidirect
47. #376C22semidirect
48. #339C22semidirect
83. C4⋊Q8D4semidirect
91. #127D4semidirect
98. #124D4semidirect
105. C4×Q8D4semidirect
109. C4×D4D4semidirect
120. #104D4semidirect
196. M16C2×D4semidirect
216. C42C2×D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#34982
#31532
#30402
C2×#4552
C2×#4532
#26572
#20042
#20022
#19652
#19632
#17592